Red-Black Tree

Introduction of Red-Black Tree

• The red-Black tree is advanced and is a modified form of the Binary Search Tree.

Definition

• A red-black tree is a self-balancing binary search tree that maintains balanced properties through the use of color annotations on its nodes i.e. every node is colored with either red or black.

Characteristics

• It ensures that the tree remains approximately balanced, guaranteeing efficient search, insertion, and deletion operations.
• When inserting or deleting a node in this tree, the tree may become temporarily unbalanced. To restore balance, several properties and rotations may be applied, which involve recoloring nodes and performing rotations. These operations ensure that the tree remains balanced while preserving the search tree property.
• It ensures that the longest path from the root to any leaf node is no more than twice the length of the shortest path. This guarantees a balanced tree structure.
• It has a guaranteed time complexity of O(log n) for basic operations like insertion, deletion, and searching which have a worst-case time complexity of O(log n) for search, insertion, and deletion operations, where n is the number of elements in the tree. The constant factors involved in these operations are typically smaller than other balanced tree structures, making red-black trees efficient in practice.

Rules & Properties

• The tree self-adjusts during insertion and deletion to preserve these properties.
• The key properties of a red-black tree are as follows:

• • Every node is either red or black.
  • The root node is always black.
  • All leaf nodes (null or empty nodes) are considered black.
  • If a node is red, both its children are black.
  • Every path from a node to its descendant leaf nodes contains the same number of black nodes. This property is known as black height.

Use/Applications

• This tree is widely used in various applications and is a fundamental data structure in many programming language libraries and databases due to its efficient performance and versatility. Some examples of their applications include implementing associative arrays, order statistics, and interval trees.
• They provide an efficient balance between search performance and tree maintenance overhead.